Applied Mathematics E-Notes, 1(2001), 139-142 \begin{matrix}c \\ ◯ \end{matrix} Available free at mirror sites of http://math2.math.nthu.edu.tw/∼amen/

New Explicit and Exact Solutions for the Nizhnik-Novikov-Vesselov Equation ^{∗}^{†}

Tie-cheng Xia^{‡}, Biao Li and Hong-qing Zhang^{§}

Received 15 July 2001

Abstract In this paper, several explicit and exact travelling wave solutions for the Nizhnik-Novikov-Vesselov equation are obtained by using the hyperbola function method and the Wu-elimination method, which include new singular solitary wave solutions and periodic solutions.

In several areas of physics such as condense matter physics [1], fluid mechanics [2], plasma physics [3] and optics [4], (1 + 1) and (2 + 1)-dimensional solitons have been studied. In this paper, we will use the hyperbola function method [5] and Wu elimination method [6] to find several new exact solutions for the (2 + 1)-dimensional Nizhnik-Novikov-Vesselov (NNV) equation [7]

u_{t} = au_{xxx} + bu_{yyy} − 3av_{x}u − 3avu_{x} − 3bω_{y}u − 3bωu_{y},  (1)
u_{x} = v_{y}, u_{y} = ω_{x},  (2)

where a and b are the arbitrary constants.

The (2 + 1)-dimensional NNV equation is an isotropic extension of the well known (1+1)-dimensional KdV equation. In order to give some new types of exact solutions of equations (1) and (2), we use hyperbola function method and Wu elimination method. By using the travelling wave transformation, equations (1) and (2) have the following formal solution:

u = φ(x, y, t) = φ(x), v = ψ(x, y, t) = ψ(ξ),  (3)
ω = θ(x, y, t), ξ = λ(x + y + kt + c).  (4)

Where λ and k are constants to be determined later and c is an arbitrary constant. Substituting equations (3) and (4) into equations (1) and (2) yields ordinary differential equations for φ, ψ, ϑ :

(a + b)λ^{2}φ′′′ − 3a(ψφ)′ − 3b(θφ)′ − kφ′ = 0,  (5)

∗Mathematics Subject Classifications: 35Q99, 35Q51, 35Q51.

†Project is supported by the National Key Basic Research Development project of China (Grant No.

1998030600) and the National Natural Science Foundation of China (Grant No. 10072031, 10072189).

‡a) Presently at Department of Applied Mathematics, Dalian University of Technology, Dalian, Liaoning 116024, P. R. China, b) Department of Mathematics, Jinzhou Normal University, Jinzhou 121000, P. R. China

§Department of Applied Mathematics, Dalian Univeristy of Technology, Dalian, Liaoning 116024, P. R. China

φ′ = ψ′,  (6)
ψ′ = θ′,  (7)

Equations (6) and (7) imply that ψ = φ + b_{0} and θ = φ + c_{0} where b_{0} and c_{0} are constants. Therefore from equations (5), (6) and (7), we have the following

λ^{2}φ′′′ − 6φφ′ − λ_{0}φ = 0,  (8)

where

λ_{0} = \frac{3ab_{0} + 3bc_{0} + k}{a + b}

Thus we only need to find φ for equation (8). According to the hyperbola function method we suppose that equation (8) has the following solution

φ = A_{0} + A_{1} sinh w + A_{2} cosh w + A_{3} sinh w cosh w + A_{4} sinh^{2} w,  (9)

and

\frac{dw}{dξ} = sinh w.  (10)

With the help of Mathematica or Maple, we have

λ^{2}φ′′′ − 6φφ′ − λ_{0}φ′
= (24λ^{2}A_{3} − 24A_{3}A_{4}) sinh^{5} w + [24λ^{2}A_{4} − 12(A_{3}^{2} + A_{4}^{2})] sinh^{4} w cosh w
+(6λ^{2}A_{2} − 18A_{1}A_{3} − 18A_{3}A_{4}) sinh^{4} w
+(6λ^{2}A_{1} − 18A_{1}A_{4} − 18A_{2}A_{3}) sinh^{3} w cosh w
+(20λ^{2}A_{3} − 12A_{0}A_{3} − 12A_{1}A_{2} − 18A_{3}A_{4} − 2λ_{0}A_{3}) sinh^{3} w
+(8λ^{2}A_{4} − 12A_{0}A_{4} − 6A_{1}^{2} − 6A_{2}^{2} − 6A_{3}^{2} − 2λ_{0}A_{4}) sinh^{2} w cosh w
+(4A_{2}λ^{2} − 6A_{0}A_{2} − 12A_{1}A_{3} − 12A_{2}A_{4} − λ_{0}A_{2}) sinh^{2} w
+(A_{1}λ^{2} − 6A_{0}A_{1} − 6A_{2}A_{3} − A_{1}λ_{0}) sinh w cosh w
+(λ^{2}A_{3} − 6A_{0}A_{3} − 6A_{1}A_{2} − λ_{0}A_{3}) sinh w
= 0.

Setting the coefficients of sinh^{j} w cosh^{i} w, i = 0, 1 and j = 0, 1, 2, 3, 4, 5 to zero, we have

24λ^{2}A_{3} − 24A_{3}A_{4} = 0,  (11)
24λ^{2}A_{4} − 12(A_{3}^{2} + A_{4}^{2}) = 0,  (12)
6λ^{2}A_{3} − 18A_{1}A_{3} − 18A_{2}A_{4} = 0,  (13)
6λ^{2}A_{1} − 18A_{1}A_{4} − 18A_{2}A_{3} = 0,  (14)
20λ^{2}A_{3} − 12A_{0}A_{3} − 12A_{1}A_{2} − 18A_{3}A_{4} − 2λ_{0}A_{3} = 0,  (15)
8λ^{2}A_{4} − 12A_{0}A_{4} − 6A_{1}^{2} − 6A_{2}^{2} − 6A_{3}^{2} − 2λ_{0}A_{4} = 0,  (16)
4A_{2}λ^{2} − 6A_{0}A_{2} − 12A_{1}A_{3} − 12A_{2}A_{4} − λ_{0}A_{2} = 0,  (17)

A_{1}λ^{2} − 6A_{0}A_{1} − 6A_{2}A_{3} − A_{1}λ_{0} = 0,  (18)
A_{3}λ^{2} − 6A_{0}A_{3} − 6A_{1}A_{2} − λ_{0}A_{3} = 0.  (19)

Using Wu’s algebraic elimination method to solve the system of overdetermined equations (11)-(19) with respect to the unknown numbers A_{0}, A_{1}, A_{2}, A_{3}, A_{4}, λ, λ_{0} yields the following cases:

Case 1: A_{0} = A_{1} = A_{2} = 0, A_{3} = ±λ^{2}, A_{4} = λ^{2}, λ_{0} = λ^{2}.

Case 2: A_{0} = A_{1} = A_{2} = A_{3} = 0, A_{4} = 2λ^{2}, λ_{0} = 4λ^{2}.

Let equations (8) and (19) combine with the following

\frac{dw}{dξ} = cosh w,  (20)

then we have Case 3: A_{0} = A_{1} = A_{3} = 0, A_{4} = 2λ^{2}, λ_{0} = 8λ^{2}. By integrating the equation dw/dξ = sinh w and taking the integration constant to

be zero, we have

sinh w = −csch ξ,  (21)

and

cosh w = − coth ξ.  (22)

Also from (20), we obtain

sinh w = − cot ξ,  (23)

and

cosh w = csc ξ.  (24)

According to equations (9), (22)–(24) and cases 1-3 we can obtain the following singular solitary wave solutions (I-II) and periodic solutions (III) for the NNV equations.

(I) u_{1}(x, y, t) = λ^{2}(csch^{2}ξ ± csch ξ coth ξ), where ξ = λ(x + y + ((a + b)λ^{2} − 3ab_{0} − 3bc_{0})t + c)

(II) u_{2}(x, y, t) = 2λ^{2}csch^{2}ξ, where ξ = λ(x + y + (4(a + b)λ^{2} − 3ab_{0} − 3bc_{0})t + c). (III) u_{3}(x, y, t) = 2λ^{2} cot^{2} ξ, where ξ = λ(x + y + (8(a + b)λ^{2} − 3ab_{0} − 3bc_{0})t + c). In summary, by using the hyperbola function method and with the aid of Mathemat-

ica and Wu-elimination method, we obtain more solitary wave solutions and periodic solutions for the NNV equation. The method can also be applied to solve other systems of nonlinear equation, such as coupled KdV equations, coupled scalar field equations and so on.

References

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